The Beltrami Equation with Parameters and Uniformization of Foliations with Hyperbolic Leaves

We consider foliations of compact complex manifolds by analytic curves. We suppose that the line bundle tangent to the foliation is negative. We show that in a generic case there exists a finitely smooth homeomophism, holomorphic on the fibers and mapping fiberwise the manifold of universal coverings over the leaves passing through some transversal $B$ onto some domain in $B\times\mathbb{C}$ with continuous boundary. depending on the leaves. The problem can be reduced to a study of the Beltrami equation with parameters on the unit disk in the case, when derivatives of the corresponding coefficient Beltrami grow no faster than some negative power of the distance to the boundary of the disk.